Relativistic Point-Coupling Model

• The relativistic point-coupling model is a covariant framework that replaces finite-range meson exchanges with zero-range contact and derivative interactions between nucleons.
• It employs self-consistent Hartree and Hartree–Bogoliubov methods to generate local scalar and vector self-energies, effectively describing ground-state properties and excitation spectra.
• Distinct parameterizations such as DD-PC1, PC-PK1, and PC-L3R enable applications across finite nuclei, hypernuclei, and dense stellar matter, highlighting its versatility.

Searching arXiv for recent and foundational papers on the relativistic point-coupling model. The relativistic point-coupling model is a covariant energy-density-functional framework in which nucleons are treated as Dirac particles interacting through local four-fermion contact terms, higher-order terms, and derivative couplings rather than explicit finite-range meson fields. Within covariant density functional theory, it is one of the two standard realizations alongside meson-exchange relativistic mean-field models. In practice, the model furnishes local scalar and vector self-energies, a relativistic Kohn–Sham or Hartree–Bogoliubov scheme for ground states, and a fully consistent source of particle–hole residual interactions for RPA-, QRPA-, and STDA-type excitation theories. Modern point-coupling functionals such as DD-PC1, DD-PCX, PC-PK1, PC-F1, and PC-L3R have been used for finite nuclei, charge-exchange response, deformation and continuum problems, hypernuclear structure, nuclear pasta, and neutron-star matter (Vale et al., 20 Sep 2025, Liu et al., 2022, Koliogiannis et al., 2024).

1. Conceptual position within covariant density functional theory

Relativistic point-coupling models replace the finite-range meson propagators of meson-exchange RMF theory by zero-range contact interactions between nucleon bilinears. The underlying building blocks are bilinears of the Dirac field,

$$\bar\psi\, O_\tau\, \Gamma\, \psi,\qquad O_\tau\in\{1,\tau_i\},$$

with Lorentz structures $\Gamma$ chosen from the standard Dirac algebra and isospin operators $1,\tau_i$. From these one constructs contact interactions in the isoscalar-scalar, isoscalar-vector, isovector-vector, and, in some realizations, isovector-scalar channels; derivative terms simulate finite-range surface effects (Vale et al., 20 Sep 2025).

This formulation preserves the characteristic advantages of covariant density functionals. The nucleon moves in a Dirac Hamiltonian, so the spin–orbit interaction arises naturally from the relativistic scalar–vector structure. In Hartree implementations, the interaction is local in coordinate space, which simplifies the residual interaction and makes point-coupling EDFs particularly convenient for configuration-space extensions, linear-response solvers, and deformed calculations (Ravlić et al., 2024, Ravlić et al., 2021).

Two broad subclasses occur in the literature represented here. One uses explicit density-dependent couplings, as in DD-PC1 and DD-PCX; the other uses density-independent leading couplings supplemented by nonlinear higher-order terms, as in PC-PK1 and PC-L3R. Both are local relativistic EDFs, but the medium dependence is encoded differently: either directly through $\alpha_i(\rho)$ or through cubic and quartic self-interactions in the densities (Liu et al., 2022, Zhang et al., 2020).

2. Covariant Lagrangian structure and effective interaction channels

A generic point-coupling Lagrangian has the schematic form

$$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$

where the omitted terms denote derivative couplings, nonlinear terms, and the electromagnetic sector (Koliogiannis et al., 2024).

For the density-dependent functional DD-PC1, the Lagrangian used in relativistic STDA applications is

$$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$

with density-dependent couplings parameterized as

$$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$

In this minimal DD-PC1 form, the isovector-scalar channel is not present explicitly (Vale et al., 20 Sep 2025).

For nonlinear point-coupling models such as PC-L3R, the Lagrangian is decomposed as

$$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$

with leading four-fermion terms

$$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$

higher-order terms

$$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$

and derivative terms in scalar, vector, and isovector-vector channels (Liu et al., 2022).

The mean fields follow from functional variation. In density-dependent point-coupling EDFs the scalar and vector self-energies take the local form

$\Gamma$0

$\Gamma$1

with rearrangement term

$\Gamma$2

These rearrangement contributions are essential for thermodynamic consistency and for a consistent derivation of the residual interaction (Ravlić et al., 2021).

3. Self-consistent mean-field, Bogoliubov, deformation, and continuum realizations

At the single-particle level, point-coupling EDFs lead to a local Dirac Hamiltonian,

$\Gamma$3

or, in fully covariant notation,

$\Gamma$4

Ground states are commonly obtained in the relativistic Hartree approximation with the no-sea prescription: negative-energy states are not occupied but are retained in the single-particle basis for excitation calculations (Vale et al., 20 Sep 2025, Liu et al., 2022).

Pairing is incorporated in relativistic Hartree–Bogoliubov theory. In finite-temperature and zero-temperature applications based on DD-PC1 or DD-PCX, the particle–hole channel is provided by the point-coupling functional, whereas the particle–particle channel is supplied by a separable Gogny-like force. The RHB equation is

$\Gamma$5

with quasiparticle occupations controlled by the grand-canonical ensemble at finite temperature (Ravlić et al., 2021).

Point-coupling models have also been implemented in deformed and continuum settings. In DRHBc theory with PC-PK1, axially deformed densities and potentials are expanded in Legendre polynomials,

$\Gamma$6

and the RHB equations are solved in a Dirac Woods–Saxon basis, allowing a unified treatment of weak binding, deformation, and continuum coupling (Zhang et al., 2020). The same framework has been extended to odd-$\Gamma$7 and odd-odd systems with blocking in the equal filling approximation and an automatic blocking procedure (Collaboration et al., 2022).

In axially deformed RHB calculations with DD-PC1 or DD-PCX, the single-particle Dirac spinors are expanded in an axially deformed harmonic-oscillator basis, and the projection $\Gamma$8 of angular momentum on the symmetry axis is conserved (Ravlić et al., 2024). This basis structure is central to later deformed pnRQRPA developments.

4. Linear response, charge exchange, and beyond-$\Gamma$9-$1,\tau_i$0 extensions

The residual interaction in point-coupling response theory is obtained from the second derivative of the EDF,

$1,\tau_i$1

Because the interaction is local, the particle–hole kernel is naturally separable in coordinate space. This feature is exploited in finite-temperature linear-response theory based on FT-RHB with DD-PC1 and DD-PCX, where the reduced Bethe–Salpeter equation is solved in channel space rather than by diagonalizing a huge QRPA matrix (Ravlić et al., 2021). For $1,\tau_i$2 shells and no explicit truncation of 2qp configurations, a conventional QRPA matrix would be about $1,\tau_i$3, whereas the separable formulation avoids that direct diagonalization (Ravlić et al., 2021).

Charge-exchange response is described by proton–neutron RQRPA or linear-response pnRQRPA built on point-coupling EDFs. In these formulations, the isovector-vector channel is supplemented by an isovector pseudovector contact term with Landau–Migdal strength $1,\tau_i$4, adjusted to Gamow–Teller data. For DD-PC1 and DD-PCX, the values $1,\tau_i$5 and $1,\tau_i$6, respectively, reproduce the GT$1,\tau_i$7 centroid in $1,\tau_i$8Pb (Ravlić et al., 2021, Ravlić et al., 2024). In a fully self-consistent DD-PCX-based PN-RQRPA, the model accurately reproduces IAR excitation energies along the Sn chain, while GT properties remain sensitive to the isoscalar $1,\tau_i$9 pairing strength (Vale et al., 2020).

The same local structure is advantageous in deformed linear response. In axially deformed pnRQRPA, the Hamiltonian is written as

$\alpha_i(\rho)$0

and the response is solved in reduced channel space. Deformation then produces pronounced fragmentation of GT and spin-dipole strength through the splitting of different $\alpha_i(\rho)$1 components, whereas the Fermi strength is almost shape-independent (Ravlić et al., 2024).

A more substantial extension beyond standard RPA/TDA is the relativistic second Tamm–Dancoff approximation. In RSTDA, the excitation operator contains $\alpha_i(\rho)$2-$\alpha_i(\rho)$3, $\alpha_i(\rho)$4-$\alpha_i(\rho)$5, and $\alpha_i(\rho)$6-$\alpha_i(\rho)$7 configurations, and the eigenvalue problem takes the block form

$\alpha_i(\rho)$8

When built on DD-PC1, RSTDA describes fragmentation and spreading of isoscalar monopole and quadrupole strength in $\alpha_i(\rho)$9O, but the unsubtracted theory exhibits cutoff dependence and infrared instabilities. The subtraction method replaces $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$0 by

$$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$1

leading to RSSTDA. In this relativistic point-coupling implementation, the inverse energy-weighted moment $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$2 is conserved between RTDA, RSSTDA(d), and RSSTDA for the complete positive spectrum, the static response satisfies $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$3, and the infrared instability is cured (Vale et al., 20 Sep 2025).

Point-coupling RPA has also been applied to superallowed $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$4 Fermi transitions. Using PC-F1 and PC-PK1, self-consistent charge-exchange RPA yields isospin-symmetry-breaking corrections $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$5 for superallowed decays and, together with experimental $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$6 values and radiative corrections, produces a first-row CKM sum that deviates from unitarity by $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$7 for all employed relativistic energy functionals (Li et al., 2011).

5. Parameterizations and finite-nucleus systematics

Several point-coupling parametrizations recur across the literature, differing mainly in how they encode density dependence and in the observables used for calibration.

Parametrization	Defining feature	Representative use
DD-PC1	Density-dependent point-coupling EDF	RSTDA in $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$8O; FT-RHB response; neutron-star and pasta studies
DD-PCX	Point-coupling EDF fitted also to charge radii and selected collective excitations	PN-RQRPA for IAR and GTR; FT charge-exchange response
PC-PK1	Nonlinear point-coupling interaction	DRHBc for even-even and odd Nd isotopes; $$\mathcal{L} = \bar\psi(i\gamma_\mu\partial^\mu-m)\psi +\sum_i \alpha_i(\rho)(\bar\psi\Gamma_i\psi)(\bar\psi\Gamma_i\psi) +\ldots,$$9-decay half-lives; single-$$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$0 hypernuclei
PC-L3R	New nonlinear point-coupling interaction optimized together with separable pairing	Global RHB ground-state systematics
PC-F1	Established nonlinear point-coupling set	Superallowed Fermi transitions; triaxial $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$1 hypernuclei

PC-L3R provides a particularly explicit benchmark for bulk performance. Fitted to the binding energies of 91 spherical nuclei, charge radii of 63 nuclei, and 12 sets of mean pairing gaps consisting of 54 nuclei in total, it yields rms$$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$2 MeV for binding energies and rms$$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$3 fm for charge radii in RHB with separable pairing. The corresponding nuclear-matter properties are $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$4, $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$5 MeV, $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$6, $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$7 MeV, $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$8 MeV, and $$\begin{aligned} \mathcal{L} =&\ \bar{\psi}\left(i\gamma_\mu\partial^\mu - m \right)\psi \ & -\frac{1}{2}\,\alpha_S(\rho)\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right) \ & -\frac{1}{2}\,\alpha_V(\rho)\left(\bar{\psi}\gamma_\mu\psi\right)\left(\bar{\psi}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\alpha_{TV}(\rho)\left(\bar{\psi}\vec{\tau}\gamma_\mu\psi\right)\left(\bar{\psi}\vec{\tau}\gamma^\mu\psi\right) \ & -\frac{1}{2}\,\delta_S\,\partial_\nu\left(\bar{\psi}\psi\right)\,\partial^\nu\left(\bar{\psi}\psi\right) \ & -e\,\bar{\psi}\gamma_\mu A^\mu \frac{1-\tau_3}{2}\psi , \end{aligned}$$9 MeV (Liu et al., 2022).

The same point-coupling architecture has been used for drip-line and continuum problems. In DRHBc calculations for even-even Nd isotopes with PC-PK1, the experimental binding energies, two-neutron separation energies, quadrupole deformations, and charge radii are reproduced rather well, while deformation and continuum coupling shift the predicted drip lines relative to spherical RCHB (Zhang et al., 2020). The odd-$$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$0 extension with automatic blocking retains comparable accuracy and supports the construction of a DRHBc mass table including odd systems (Collaboration et al., 2022).

Weak-interaction applications form another major branch. In neutron-rich even-even nuclei with $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$1, a self-consistent pn-QRPA built on PC-PK1 shows that isoscalar proton–neutron pairing can significantly reduce $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$2-decay half-lives, and with an isospin-dependent isoscalar pairing strength the calculated half-lives reproduce experiment well (Wang et al., 2015). A plausible implication is that the local, covariant point-coupling structure is not restricted to static observables but supports quantitatively competitive spin–isospin dynamics across a large region of the nuclear chart.

6. Dense matter, nuclear pasta, and neutron stars

Point-coupling EDFs have also been carried into uniform and nonuniform matter. In one approach, the baryonic energy density in $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$3-equilibrated matter is expanded as

$$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$4

with

$$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$5

Within DD-PC families constrained by finite nuclei, CREX/PREX-2 observables, and GW170817, exponential correlations are found between finite-nucleus quantities such as $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$6 or neutron-skin thickness and neutron-star quantities $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$7 and $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$8. Including the fourth-order symmetry energy $$\alpha_i(x)=a_i+\left(b_i+c_i x\right)e^{-d_i x},\qquad x=\rho/\rho_{\text{sat}} .$$9 generally yields larger radii than truncation at $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$0 alone, but the inferred constraints from CREX and PREX-2 remain mutually inconsistent within the DD-PC framework (Koliogiannis et al., 2024).

A complementary three-dimensional Thomas–Fermi program based on point-coupling RMF describes low-density inhomogeneous matter and nuclear pasta. The Lagrangian includes an additional mixed isoscalar-vector–isovector-vector term proportional to $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$1, which acts as the point-coupling analogue of the $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$2–$$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$3 coupling and is used to tune the symmetry-energy slope $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$4. For fixed proton fraction, the model produces droplets, rods, slabs, tubes, and bubbles, together with some intermediate pasta structures at relatively large proton fraction. In cold $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$5-equilibrated stellar matter, however, the proton fraction is so small that nonspherical pasta shapes are unlikely in neutron-star crusts (Ji et al., 2021).

Direct neutron-star applications have exposed the sensitivity of point-coupling parametrizations to poorly constrained high-density behavior. Using DD-PC1 and PC-PK1 as examples, the equation of state of neutron-star matter yields markedly different proton fractions, core–crust transition densities, crustal moments of inertia, and tidal deformabilities. DD-PC1 remains monotonic in pressure and gives smaller radii and smaller $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$6, whereas PC-PK1 exhibits a suppression of pressure at high densities caused by its quartic vector term, making predictions near the maximum mass difficult (Sun et al., 2019). This suggests that finite-nucleus fits alone do not adequately constrain the suprasaturation isoscalar and isovector sectors.

7. Hypernuclei, impurity effects, and current limitations

The point-coupling formalism extends naturally to hypernuclei by adding $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$7 contact couplings. In the single-$$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$8 extension, the Lagrangian supplements the nucleonic point-coupling sector with scalar and vector $$\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm 4f}+\mathcal{L}^{\rm ho}+\mathcal{L}^{\delta}+\mathcal{L}^{\rm em},$$9 terms, derivative couplings, and a tensor term,

$$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$0

including

$$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$1

gradient terms, and

$$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$2

Fitted interactions reproduce $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$3 binding energies over a wide mass region, while some parameter sets yield reverse ordering of the spin–orbit partners in heavy hypernuclei. The tensor coupling is central because it suppresses the $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$4 spin–orbit splitting while keeping the scalar and vector $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$5 couplings close to quark-model expectations (Tanimura et al., 2011).

The same point-coupling logic underlies triaxial and beyond-mean-field studies of hypernuclear collectivity. Using PC-F1 for the nucleonic EDF and PCY-S1 for the $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$6 sector, triaxial RMF calculations and a five-dimensional collective Hamiltonian show that a $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$7 in the lowest positive-parity state tends to reduce deformation and collectivity, whereas a $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$8 in a negative-parity $$\begin{aligned} \mathcal{L}^{\rm 4f} &= -\frac{1}{2}\alpha_S(\bar\psi\psi)^2 -\frac{1}{2}\alpha_V(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) \ &\quad -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi), \end{aligned}$$9 orbit tends to favor larger deformation and can alter oblate–prolate competition (Xue et al., 2014). In a microscopic particle-rotor model with relativistic point-coupling $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$0 interaction, the derivative and tensor terms reduce the $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$1 binding energy as their strengths increase; specifically, the tensor term decreases the energy splitting between the first $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$2 and $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$3 states and increases the energy splitting between the first $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$4 and $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$5 states in $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$6 (Mei et al., 2016).

Several limitations recur across the literature. Hartree implementations omit Fock terms; some applications use pure Hartree ground states with no explicit ground-state correlations beyond mean field (Vale et al., 20 Sep 2025). Spin–isospin sectors often require additional couplings such as $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$7 or $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$8 pairing strengths that are not fixed by the ground-state fit (Vale et al., 2020, Ravlić et al., 2021). Deformed pnRQRPA still omits continuum RQRPA in axial geometry and beyond-QRPA correlations such as quasiparticle–phonon coupling (Ravlić et al., 2024). In RSTDA, the present configuration space neglects $$\mathcal{L}^{\rm ho} = -\frac{1}{3}\beta_S(\bar\psi\psi)^3 -\frac{1}{4}\gamma_V\big[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)\big]^2 -\frac{1}{4}\gamma_S(\bar\psi\psi)^4,$$9-$\Gamma$00-$\Gamma$01 and $\Gamma$02-$\Gamma$03 components (Vale et al., 20 Sep 2025). For neutron-star matter, the large divergence between DD-PC1 and PC-PK1 indicates that high-density constraints from tidal deformabilities, masses, and radii are essential for future parameterizations (Sun et al., 2019).

Taken together, these developments define the relativistic point-coupling model as a broad, internally consistent covariant EDF methodology rather than a single parametrization: a local Dirac-based functional framework whose zero-range interaction structure supports self-consistent treatments of finite nuclei, collective excitations, hypernuclei, and dense matter, while leaving open a continuing program of improvement in the isovector, spin–isospin, and high-density sectors.